Depth and Stanley Depth of the Quotient Ring of Edge Ideals Associated with Some Classes of Lobster Trees and Unicyclic Graphs

Abstract

In this Thesis, we discuss rings and modules, which are the fundamental algebraic structures of Abstract Algebra. Moreover, we discuss two algebraic invariants of a module which are Stanley depth and depth of a module. In addition, we discuss how the regular element property helps in determining the depth of a module. Then, we discuss some recent results of depth and Stanley depth of edge ideals associated with di erent graphs. Thereafter, we nd the Stanley depth of some modules and edge ideals using the method of posets. Lastly, we compute the Stanley depth and depth of the quotient ring of edge ideals associated with di erent classes of graphs. These classes include some lobster trees and unicyclic graphs. Then, we show that the values of depth and Stanley depth are equal and can be stated in terms of n and m. In addition, we prove the Stanley's inequality for modules associated with these classes of graphs.